the aspen–amsterdam void finder comparison project

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The Aspen–Amsterdam Void Finder Comparison Project

Jorg M. Colberg1,2, Frazer Pearce3, Caroline Foster4,5, Erwin Platen6,

Riccardo Brunino3, Mark Neyrinck7, Spyros Basilakos8, Anthony Fairall9,

Hume Feldman10,Stefan Gottlober11, Oliver Hahn12, Fiona Hoyle13,

Volker Muller11, Lorne Nelson4, Manolis Plionis14,15, Cristiano Porciani12,

Sergei Shandarin10, Michael S. Vogeley16, Rien van de Weygaert6

1 Carnegie–Mellon University, 5000 Forbes Ave, Pittsburgh, PA 15213, USA2 Astronomy Department, University of Massachusetts, Amherst, MA 01003, USA3 School of Physics & Astronomy, University of Nottingham, Nottingham, NG7 2LE, UK4 Bishop’s University, Department of Physics, 2600 College Street, Sherbrooke, QC J1M 0C8, Canada5 Centre for Astrophysics & Supercomputing, Swinburne University of Technology, Hawthorn VIC 3122, Australia6 Kapteyn Institute, University of Groningen, PO Box 800, 9700 AV Groningen, The Netherlands7 Institute for Astronomy, University of Hawaii, Honolulu, HI 96822, USA8 Institute of Astronomy & Astrophysics, National Observatory of Athens, I. Metaxa & B. Pavlou, P. Penteli 152 36, Athens, Greece9 Department of Astronomy, University of Cape Town, Private Bag, Rondebosch 7700, South Africa10 Department of Physics and Astronomy, University of Kansas, Lawrence, KS 66045, USA11 Astrophysikalisches Institut Potsdam, An der Sternwarte 16, 14482 Potsdam, Germany12 ETH Zurich, 8093 Zurich, Switzerland13 Department of Physics and Astronomy, Widener University, One University Place, Chester, PA 1901314 Institute of Astronomy & Astrophysics, National Observatory of Athens, I.Metaxa & B.Pavlou, Palaia Penteli, 152 36 Athens, Greece15 Instituto Nacional de Astrofisica, Optica y Electronica (INAOE) Apartado Postal 51 y 216, 72000 Puebla, Pue., Mexico16 Department of Physics, Drexel University, 3141 Chestnut Street, Philadelphia, PA 19104, USA

Accepted 200? ???? ??. Received 2007 ???? ??; in original form 2007 xx

ABSTRACT

Despite a history that dates back at least a quarter of a century studies of voids inthe large–scale structure of the Universe are bedevilled by a major problem: there exista large number of quite different void–finding algorithms, a fact that has so far gotin the way of groups comparing their results without worrying about whether such acomparison in fact makes sense. Because of the recent increased interest in voids, bothin very large galaxy surveys and in detailed simulations of cosmic structure formation,this situation is very unfortunate. We here present the first systematic comparisonstudy of thirteen different void finders constructed using particles, haloes, and semi–analytical model galaxies extracted from a subvolume of the Millennium simulation.The study includes many groups that have studied voids over the past decade. Weshow their results and discuss their differences and agreements. As it turns out, thebasic results of the various methods agree very well with each other in that they alllocate a major void near the centre of our volume. Voids have very underdense centres,reaching below 10 percent of the mean cosmic density. In addition, those void findersthat allow for void galaxies show that those galaxies follow similar trends. For example,the overdensity of void galaxies brighter than mB = −20 is found to be smaller thanabout −0.8 by all our void finding algorithms.

Key words: cosmology: theory, methods: N-body simulations, dark matter, large-scale structure of Universe

1 INTRODUCTION

Large regions of space that are only sparsely populated withgalaxies, so–called voids, have been known as a feature of

galaxy surveys since the first of those surveys was compiled,the most well–known cases being the famous void in Bootes,discovered by Kirshner et al. (1981), and the first void sam-

http://arXiv.org/abs/0803.0918v1

2 J. M. Colberg et al.

ple of de Lapparent, Geller & Huchra (1986). However, dueto the fact that voids occupy a large fraction of space, onlyrecently have galaxy surveys become large enough to allowsystematic studies of voids and the galaxies inside them.For recent studies of voids and void galaxies in the two–degree field Galaxy Survey (2dFGRS, Colless et al. 2001)and the Sloan Digital Sky Survey (SDSS, York et al. 2000)see Hoyle & Vogeley (2004), Ceccarelli et al. (2006), Patiriet al. (2006a), Tikhonov (2006), von Benda-Beckmann &Muller (2008) and Rojas et al. (2004), Goldberg et al. (2005),Hoyle et al. (2005), Rojas et al. (2005), Patiri et al. (2006b),respectively. Also see Croton et al. (2004) for a recent, de-tailed study of the void probability function in the 2dF.

On the theoretical side, progress has been mirrored byvast improvements in models and simulations, with system-atic studies of large numbers of voids now being common(see, for example, the early works by Regos & Geller 1991,Dubinski et al. 1993 or Van de Weygaert & Van Kampen1993, and the more recent Arbabi-Bidgoli & Muller 2002,Mathis & White 2002, Benson et al. 2003, Gottlober et al.2003, Goldberg & Vogeley 2004, Sheth & Van de Weygaert2004, Bolejko et al. 2005, Colberg et al. 2005, Padilla et al.2005, Brunino et al. 2006, Furlanetto & Piran 2006, Hoeftet al. 2006, Lee & Park 2006, Park & Lee 2006, Patiri etal. 2006c, Shandarin et al. 2006). Theory shows that voidsare a real feature of large–scale structure, since initially un-derdense regions grow in size as overdense regions collapseunder their own gravity (see, for example, Sheth & Van deWeygaert 2004). But while the general picture appears tobe well supported by the standard Λ Cold Dark Matter(ΛCDM) cosmology, Peebles (2001) pointed out some po-tentially critical issues. Does the ΛCDM cosmology producetoo many objects in voids that have no observational coun-terparts? A detailed discussion of other reasons why studiesof voids are an interesting topic is beyond the scope of thispaper. Briefly, their role as a prominent feature of the Mega-parsec Universe means that a proper and full understandingof the formation and dynamics of the Cosmic Web is not pos-sible without probing the structure and evolution of voids.A second rationale is that of inferring global cosmologicalinformation from the structure and geometry of and outflowfrom voids. The third aspect is that of providing a uniqueand still largely pristine environment for testing theories ofthe formation and evolution of galaxies.

Despite the growing interest in voids and the large num-ber of recent studies, a fairly significant problem remains:as it turns out, almost every study uses its own void finder.There is general agreement that there are voids in the dataor in the simulations, but many different ways were proposedto find them. Thus, the resulting voids are either spherical(with or without overlap), shaped like lumpy potatoes1, orthey percolate all across the studied volume. What is more,some groups do not allow for the existence of void galax-ies, whereas many others do. An added complication is thatmany theoretical studies use the dark matter distributionto find voids, whereas observational studies have to rely ongalaxies. As a consequence, it is not clear how the resultsfrom studies done by different groups can be compared, es-

1 JMC admits that this picture, while being accurate, is not verypretty.

pecially if observational and theoretical results are broughttogether. What most studies so far can agree on is thata) voids are very underdense in their centres (approachingaround five percent of the mean density) and that b) voidsoften have very steep edges. In other words, the number ofboth observed and simulated galaxies increases very rapidlywhen reaching the edge of a void, and the corresponding re-sult has been found for the density of dark matter in studiesthat used dark–matter only simulations.

Given the disagreements in the different methods, whichare in part due to the different nature of the data sets used,the aim of this work is very modest. We apply thirteen dif-ferent void finders, all of which have been used over thepast decade to study voids, to the same data set in or-der to compare the results. As our data set we use parti-cles, haloes, and semi–analytical model galaxies (Croton etal. 2005) from a subvolume of the Millennium simulation(Springel et al. 2005) specifically selected to be underdenseand therefore void-rich. That way, while the methods areas different as finding connected cells on a density grid andidentifying empty regions in the model galaxy distributionby eye, a meaningful comparison is still possible, since eachvoid finder treats a subset of the same data set.

The aim of this paper is not to argue which void finderprovides the best way to identify voids. We do hope, how-ever, that this paper will allow the reader to understand thedifferences between the different void finders so that it willbe easier to compare different studies of voids in the litera-ture. We also hope that this paper will trigger more detailedfollow–up studies to work towards a more unified view of thistopic and to study properties of voids not covered here, suchas for example their shapes and orientations, in detail.

This paper is organised as follows: in Section 2 we detailthe simulation from which the test region was extracted anddescribe the procedure each group was asked to undertake.In Section 3 we briefly describe each void finding algorithm,before we undertake a comparison of the voids found in Sec-tion 4. Section 5 contains a summary and conclusions.

2 THE SIMULATION AND EXTRACTION

PROCEDURE

For this work, we use the Millennium simulation (Springelet al. 2005) and a matched z = 0 galaxy catalogue, cre-ated using a semi–analytical galaxy formation model (Cro-ton et al. 2005). The simulation of the concordance ΛCDMcosmology contains 21603 particles in a (periodic) box ofsize 500 h−1 Mpc in each dimension. The cosmological pa-rameters are total matter density Ωm = 0.25, dark en-ergy/cosmological constant ΩΛ = 0.75, Hubble constanth = 0.73, and the normalisation of the power spectrumσ8 = 0.9. With these parameters, each dark matter particlehas a mass of 8.6 × 108 h−1 M⊙.

In the Millennium simulation volume we located a60h−1 Mpc region centred on a large void and extractedthe coordinates of the 12,528,667 dark matter particles con-tained within it. This subvolume thus has a mean densitywhich is lower than the cosmic mean, corresponding to anoverdensity δ = ρ/ρ − 1 = −0.28.

We also extracted a list of the 17,604 galaxies togetherwith their BVRIK dust corrected magnitudes (down to B=-

Void Finder Comparison Project 3

10) that are present in the semianalytic catalogue of Crotonet al. (2005) within this volume and the 4,006 dark mat-ter halos present in the subfind catalogue (a clean spheri-cal overdensity based catalogue) with masses greater than1011 h−1 M⊙. Note that while the small volume prohibitsstatistical comparisons between void finders it allows forvoid–by–void comparisons.

Each group was asked to run their void finder with theirpreferred parameters on this database and return a void listfor the voids found, tagging each of the dark matter parti-cles, galaxies, and haloes with the void identifier of the voidthey resided in. This allowed simple plotting and analysisof each void sample. For overlapping voids, the dark matterparticle was to be assigned to the larger void. As the regionis not periodic we only requested information about voidswhose centres lay within the central 40 h−1 Mpc region (i.e.the outer 10 h−1 Mpc was to be neglected).

The top left and top centre panels of Figure 1 show slicesof thickness 5 h−1 Mpc through this central region. The topleft panel only contains the distribution of the dark matter,whereas the top centre panel includes model galaxies on topof the dark matter distribution. The largest halo has a massof only 1.75×1012 h−1 M⊙, so filaments in these images cor-respond only to the less massive filaments in standard slicesthrough the dark matter distribution as seen in, for exam-ple, Springel et al. (2005). Furthermore, the slice contains atotal of 145,194 dark matter particles, equivalent to an over-density of δ = −0.77. It is important to keep these numbersin mind when studying the results obtained by the variousvoid finders. Subsequent panels show the largest void iden-tified by each group and those galaxies contained within allvoids identified.

3 VOID FINDERS

This section gives a brief outline of the void finders used forthis study, grouped into those which rely on the dark matterdistribution and those which rely on the sparser galaxy orhalo distributions (also see Table 1 for a general overview).For more details, the interested reader is referred to theindividual studies by the different groups. Anyone simplyinterested in the results can skip to the next section. Pleasenote that in this study, all group finders use real–space data.

3.1 Finders based on the dark matter distribution

3.1.1 Colberg: Irregularly shaped underdense regionsaround local density minima

This method was introduced in Colberg et al. (2005), whereit was used to study voids in the dark matter distributionof a suite of large N–body simulations. The starting pointfor Colberg et al.’s void finder is the adaptively smootheddistribution of the full dark matter distribution in a simula-tion. Proto–voids are constructed in a fashion quite similarto Hoyle & Vogeley’s void finder, the difference being thatColberg’s uses local minima in the density field as the centresof voids, and the mean density of the spherical proto–voidsis required to be smaller or equal to an input threshold,which, following a simple linear theory argument, is takento be δ = −0.8 (Blumenthal et al. 1992). Proto–voids are

then merged according to a set of criteria, which allow forthe construction of voids that can have any shape, as long astwo large regions are not connected by a thin tunnel (whichwould make the final void look like a dumbell). The voidsthus can have arbitrary shapes, but they typically look likelumpy potatoes.

For this study, a grid of size 4803 and a minimum voidradius of rmin = 2.0 h−1 Mpc were used. In the following,this void finder and its results are referred to as Colberg.

3.1.2 Pearce: Spheres around local density minima

For every particle in the Millennium simulation local densi-ties were calculated by smoothing over the nearest 32 neigh-bours using a beta spline kernel (Monaghan & Lattanzio1985). This list was then ranked in density order (start-ing from the most underdense particle), and independentinitial void centres were chosen such that they were morethen 2h−1 Mpc away from a previously selected centre (upto δ = −0.965). The radial distribution of particles aboutthese trial centres was then used to calculate the first up–crossing above δ = −0.9. These radii were then sorted insize order, and the resulting list was cleaned by removingvoids whose centre lay within an already found void. The3,024 voids found by this procedure were used as the start-ing points for the more traditional halo based group finderused by Brunino et al. (2007). In the following, this voidfinder and its results are referred to as Pearce.

3.1.3 Hahn/Porciani: Equation of motion in smootheddensity field

A stability criterion for test–particle orbits is used to dis-criminate four environments with different dynamics (Hahnet al. 2006). The classification scheme is based on a seriesexpansion of the equation of motion for a test particle in thesmoothed matter distribution. The series expansion yields azero order term, the acceleration, and a second order term,the tidal field Tij (Hessian of the potential). The eigenvaluesof Tij characterise the triaxial deformation of an infinites-imal sphere due to the gravitational forces. Voids are clas-sified as those regions of space where Tij has no positiveeigenvalues (tidally unstable).

The method has one free parameter, the size of theGaussian filter used to smooth the potential. This parame-ter is set to Rs = 2.09 h−1Mpc, which corresponds to a massof 1013 h−1M⊙ contained in the filter at mean density. Thischoice gives excellent agreement with a visual classification(see the discussion in Hahn et al. 2006). This mass scalecorresponds to about 2M∗ at z = 0.

The tidal field eigenvalues are evaluated on a grid. Then,contiguous regions classified as voids are linked together.Voids can thus have arbitrary shapes, and their volumes areproportional to the number of cells linked together. In thefollowing, this void finder and its results are referred to asHahn/Porciani.


Author Base Method

Brunino Haloes Spherical regions in halo distributionColberg Dark matter density field Irregularly shaped underdense regions around local density minimaFairall Galaxies Empty regions in galaxy distributionFoster/Nelson Galaxies Empty regions in galaxy distributionGottlober Haloes/Galaxies Spherical empty regions in point setHahn/Porciani Dark matter density field Tidal instability in smoothed density fieldHoyle/Vogeley Galaxies Empty regions in galaxy distributionMuller Halos/Galaxies Empty convex regions in point setNeyrinck Dark matter density field ZOBOV, depressions in unsmoothed DM fieldPearce Dark matter Local density minima spheresPlaten/Weygaert Dark matter density field Watershed DTFEPlionis/Basilakos Dark matter density field Connected underdense density grid cellsShandarin/Feldman Dark matter density field Connected underdense density grid cells

Table 1. An overview of the void finders used in this study.

3.1.4 Neyrinck: Zobov

ZOBOV (ZOnes Bordering On Voidness, Neyrinck (2008))is an inversion of a publicly available halo-finder, VOBOZ2

(Neyrinck et al. 2005). ZOBOV differs from VOBOZ in thatZOBOV looks for density minima instead of maxima, anddoes not consider gravitational binding. ZOBOV has someunique features: it is entirely parameter-free, working di-rectly on the unsmoothed particle distribution; and, it re-turns all (even possibly spurious) depressions surroundingdensity minima, along with estimates of the probability thateach arises from Poisson noise.

The first step is density estimation and neighbouridentification for each dark-matter particle, using whatSchaap (2007) calls the Voronoi Tesselation Field Estima-tor. ZOBOV then partitions the particles into zones (de-pressions) around each minimum. Each particle jumps to itslowest-density neighbour, repeating until it reaches a mini-mum. A minimum’s zone is the set of particles which flowdownward into it. Zones resemble voids, but because of un-smoothed discreteness noise, many zones are spurious, andothers are only cores of voids detected by eye. So, ZOBOVmust join some zones together to form voids. Voids aroundeach zone grow by analogy with a flooding landscape (rep-resenting the density field): water flows into neighbouringzones, adding them to the original zone’s void. The zone’svoid stops growing when the water spills into a zone deeperthan the original zone, or the whole field is submerged. Theprobability that a void is real is judged by the ratio of thedensity at which this happens to the void’s minimum den-sity.

This density contrast r is converted to a probabilitythrough comparison with a Poisson point distribution; seeNeyrinck et al. (2005) for details. The ZOBOV catalogueused for comparison with other void-finders includes onlyvoids exceeding a 5-σ probability threshold, which corre-sponds to a density contrast of 2.89. Also, subvoids exceed-ing this threshold have been removed from parent voids. Inthe following, this void finder and its results are referred toas Neyrinck.

2 Available at http://ifa.hawaii.edu/∼neyrinck/VOBOZ.

3.1.5 Platen/Weygaert: Watershed void finder

The Watershed Void Finder (WVF) is an implementationof the Watershed Transform (WST) for image segmentationtowards the analysis of the Cosmic Web. The WatershedTransform is a familiar concept in mathematical morphologyand was first introduced by Beucher & Lantuejoul (1979,also see Beucher & Meyer 1993).

The WST delineates the boundaries of separate do-mains, ie. the basins, into which which the yields of e.g.rainfall will collect. The analogy with the cosmological con-text is straightforward: voids are to be identified with thebasins, while the filaments and walls of the cosmic web arethe ridges separating the voids from each other.

The voids are computed by an algorithm that mimicsthe flooding process. First the cosmological point distribu-tion is transformed by the DTFE technique into a densityfield. DTFE (Schaap & van de Weygaert 2000, Schaap 2007)assures an optimal rendering of the hierarchical, anisotropicand voidlike nature and aspects of the weblike cosmic mat-ter distribution. The density field is adaptively smoothedby nearest neighbour median filtering (Platen et al. 2007).Minima are selected from the smoothed field and marked asthe sources of flooding. While the ”watershed” level rises agrowing fraction of the ”landscape” will be flood: the basinsexpand. Ultimately basins will meet at the ridges, saddle-points in the density field. These ridges define their bound-aries, and are marked as edge separating the two basins.The procedure is continued until the density field is com-pletely immersed, leaving a division of the landscape intoindividual segments separated by edges. The edges delineatethe skeleton of the field and outline the voids in the densityfield.

The voids in the watershed procedure have no shapeconstraints. By definition the voids fill space completely.Nearly without exception galaxies and dark halos are lo-cated on the ridges of the cosmic web, implying a minimalamount of galaxies to be located in the watershed void seg-ments.


3.1.6 Plionis/Basilakos: Connected underdense densitygrid cells

This void finder is applied on a regular 3-D grid of theDM particle distribution or of a smoothed galaxy distribu-tion, and it is based in identifying those grid cells (whichwe call “void cells”) whose density contrast lies below aspecific threshold. Then all neighbouring (touching) “voidcells” are connected to form candidate voids (see also Plio-nis & Basilakos 2002). Therefore, by construction, voids donot overlap, and they can have an arbitrary shape, whichis approximated by an ellipsoidal configuration (see Plionis& Basilakos 2002). Of course, increasing the threshold onetends to percolate through the available volume by connect-ing voids. The threshold below which the “void cells” areidentified is chosen so that a specific fraction of the proba-bility density function (pdf) is used. For example, the voidspresented here are based on the lowest 12.5% density “voidcells”, which corresponds to δρ/ρ ≃ −0.92.

In order to identify significant voids from our candidatelist we compare with voids found in 1000 realizations of theDM particle distribution, using again the lowest 12.5% den-sity void cells of each “random–realization” pdf. Now a prob-ability curve as a function of void size can be built. Smallervoids appear with a large frequency in the random realiza-tions and thus a candidate void is considered as significantonly if its probability of appearing in a random distributionis < 0.05.

There are two free parameters in this void identificationprocedure: (a) The grid cell size and (b) the threshold belowwhich void cells are identified. The first is selected arbitrarilyin this work such that it roughly encloses the volume ofa typical cluster of galaxies, (2Mpc)3, while the second isselected such that it maximises the number of significantvoids. In the following, this void finder and its results arereferred to as Plionis/Basilakos.

3.1.7 Shandarin/Feldman: Connected underdense densitygrid cells

Voids are defined as the individual 3D regions of the low-density excursion set fully enclosed with the isodensity sur-faces (for more details see Shandarin et al. 2006). Here, wefirst generate the density field on a uniform rectangular gridusing the cloud-in-cell (CIC) technique. The grid parameteris chosen to be equal to the mean separation of particlesin the whole simulation d = (500/2160)h−1 Mpc. The CICalgorithm uses particles of the same size. Then the den-sity field is smoothed with a spherical Gaussian filter withRG = 1h−1 Mpc, assuming nonperiodic boundary condi-tions and empty space beyond the boundaries. In the anal-ysis we use only the central part of the cube, slicing 4.5dfrom every face of the initial cube affected by smoothing.The final cube consists of 2503 grid sites with the volume ofabout 91% of the initial cube. Nonpercolating voids reachmaximum sizes at the percolation transition (Shandarin etal. 2004). The technique makes no assumptions about theshapes of voids that generally are highly nonspherical. Athigher thresholds the total volume in all but the percolatingvoid drops off precipitously and the excursion set practicallybecomes a single percolating void. Our voids are identifiedat the percolation threshold δ ≈ −0.88 (filling fraction of

the voids, FFV = 20%). We find 19 voids larger than 5h−3

Mpc3. The largest void is of irregular shape and its vol-ume is V = 2.1× 104h−3 Mpc3. There are neither halos norgalaxies inside these voids. Galaxies start to appear in thepercolating void at δ > −0.86 (FFV >27% ) and halos atδ > −0.63 (FFV >66%). In the following, this void finderand its results are referred to as Shandarin/Feldman.

3.2 Finders based on the galaxy or halo

distribution

3.2.1 Brunino: Spherical voids in halo catalogue

This void finder algorithm uses the void centres provided byPearce’s algorithm as an initial guess for the location of theunderdense regions. These positions are then used to searchfor the maximum spheres that are empty of haloes withmasses larger than 8.6×1011h−1M⊙, or 1000 particles. Tocharacterise the final position of the void centres and theirradii we populate a sphere of radius R = 5h−1 Mpc,centredon each initial position, with 2000 random points (the choiceof these quantities has proved to be the most convenientin order to obtain a stable result). For every point in thissphere, the position of the closest four haloes lying in geo-metrically “independent” octants is found. The sphere de-fined by these four haloes is then built. This is repeated forall the 2000 random points. As a characterisation (positionand radius) of the void, the biggest empty spherical regiongenerated in the previous step is chosen.

It is important to stress that the position of the voiddefined in this way normally does not match the positionof the initial guess. Furthermore, in the present work, voidswhose centre turned out to lie inside a larger void have beendiscarded. A total of six void regions have been found inthe volume of interest, three of which have been neglectedapplying this criteria. This algorithm has been developed toresemble the observational technique used to detect voidsto enable a more direct comparison with simulations (e.g.Trujillo et al. 2006, Brunino et al. 2007) In the following,this void finder and its results are referred to as Brunino.

3.2.2 Fairall: Voids in the galaxy distribution

Voids have been located manually by inspection of slicevisualisations: A moving slice, in x and y with thickness∆z = 5 h−1 Mpc, has been passed through the data in stepsof 2.5 h−1 Mpc. Its progress has been visualised by softwarethat shows both individual galaxies and large–scale struc-tures, the latter based on minimal spanning trees with perco-lations of 1 h−1 Mpc or less (effectively “friends of friends”).The voids are conspicuous cavities, approximately spheri-cal, empty or almost empty of galaxies, visible in consecu-tive slices, with sharply defined walls formed by large–scalestructures. Since the voids interconnect with one another,the large–scale structures do not necessarily completely en-close each void. If a void departs from sphericity, an averageradius is estimated. Where the data allow, distinct voids assmall as 2.5 h−1 Mpc (radius) are identified. In the following,this void finder and its results are referred to as Fairall.


3.2.3 Foster/Nelson: Voids in the galaxy distribution

The identification of voids is calculated using a prescrip-tion similar to that of Hoyle and Vogeley (2002). The algo-rithm has been extensively employed to analyse void struc-ture and distribution using the results from the recently pub-lished Data Release 5 (DR5) of the Sloan Digital Sky Survey(SDSS) (see, Foster and Nelson 2007). The average distanceto the third nearest neighbour (d) in the sample and itsstandard deviation (σ) are calculated. In order to ensure ahigh degree of confidence in identifying bona fide voids, weuse the parameter R3 = d + λ σ to distinguish wall galaxiesfrom field galaxies and set λ = 2. Wall galaxies are defined asthose galaxies whose third nearest neighbour is closer thanR3. All other galaxies are field galaxies. The wall galaxies areplaced in a grid whose basic cell geometry is cubic having aside of length R3/2. The empty cells are then identified andeach empty cell acts as a seed from which holes are grown.A hole is defined as a sphere that is entirely devoid of wallgalaxies. Its radius and centre are computed such that thereare exactly 3 wall galaxies on its surface. Voids are thenformed by amalgamating the overlapping holes starting withthe largest holes. Only holes whose radius exceeds a certainthreshold value (Rmin = 7.5 h−1 Mpc for this analysis) canform voids; those that are smaller are used to map out theboundary surface of a pre–existing void. Thus if there areno holes whose size exceeds the threshold, no voids will beidentified. The position of the centre of each void is calcu-lated by finding the ”’centre of volume”’. The position ofthe centre and the volume are calculated using Monte Carlomethods and the equivalent spherical radius is determined.In the following, this void finder and its results are referredto as Foster/Nelson.

3.2.4 Gottlober: Empty spheres in point set

The void finder starts with a selection of point–like objects inthree–dimensional space. These objects can be halos abovesome mass (or circular velocity) or galaxies above some lu-minosity. Thus voids are characterised by the threshold massor luminosity.

For the data used here, Ng = 380 grid cells in eachdimension were used, which corresponds to a grid cell sizeof 158 h−1 kpc. On this grid the point is found, which hasthe largest distance to the set of points defined above. Thisgrid point is the centre of the largest void. This void is thenexcluded, and the procedure is repeated by searching fora point with the largest distance to the set. Iterating thisprocedure thus yields the full sample of voids

In principle, the algorithm allows to have a certain num-ber of points (objects above the threshold mass or luminos-ity) inside the void. Here, this number is set to zero, i.e.the voids are completely empty with respect to the definedsample. Of course, they may contain objects with smallermasses or lower luminosities than the assumed threshold.

In principle, the algorithm allows for the construction ofvoids with arbitrary shape. The starting point is the spher-ical void described above. It can be extended by spheres oflower radius which grow from the surface of the void into allpossible directions. However, in this test case this featurewas switched off, and the search was restricted to sphericalvoids to avoid ambiguities of the definition of allowed devia-

tions from spherical shape. In the following, this void finderand its results are referred to as Gottlober.

3.2.5 Hoyle/Vogeley: Voidfinder

Voidfinder was introduced in Hoyle & Vogeley (2002; HV02)and has been used frequently to locate voids in galaxy sur-veys (Hoyle & Vogeley 2004, Hoyle et al. 2005). Full detailsof how voidfinder works can be found in Hoyle & Vogeley(2002), so here we will only briefly summarise the algorithm.Voidfinder operates on samples of galaxies and is based onthe ideas discussed in El–Ad & Piran (1997) and El–Ad et al.(1997). In a volume–limited galaxy catalogue (with a typicallimit just fainter than M*), galaxies are first pre–categorisedinto wall or void galaxies, depending on the distances to thegalaxies’ third–nearest neighbours. Wall galaxies are thenbinned into cells of a cubic grid. Around the centres of allempty grid cells the largest possible spheres that are alsoempty are found. Finally, the set of unique voids is con-structed by determining maximal spheres and their over-laps. voidfinder voids are non–spherical. A minimum voidsize of 10 h−1 Mpc is set to only select the largest, statisti-cally most significant voids. For tests of voidfinder using sim-ulation data see Benson et al. (2003). In the following, thisvoid finder and its results are referred to as Hoyle/Vogeley.

3.2.6 Muller: Empty convex regions in point set

This grid based void finder looks first for empty base voidsin the halo/galaxy sample, and then it adds extensions toapproximate spherical voids. It was run first with only abase void search and then with extensions. The idea of thevoid finder is to look for empty nearly convex regions inthe galaxy distribution. It is based on a grid on the surveyvolume where cells with galaxies are marked as occupied. Inthe next step, it looks for maximum cubes on the grid thatare empty of galaxies and previously found voids. We callthis a base voids, and get a first catalogue of cube voids,the most simple algorithm, but it produces a void cataloguewith similar sizes as assuming a spherical base volume. Aslightly more refined method is to extend the base voidsalong the faces with adding square sheets empty of galaxiesand not contained in previously found voids. This extensionprocedure is iterated. To avoid extended fingers or bridgesbetween voids, we require the extension to have a surfacebigger than 2/3 (an arbitrary parameter) of the previousone. These extended voids have on average the same volumein the extensions as in the base voids. We measuring the sizeby the effective cube size, i.e. a cube of the same volume asthe base plus extensions. Such voids are in general largerthan selecting square voids. This void finder is based onthe prescription of Kauffmann & Fairall (1991), which wasfurther developed and tested by Muller et al. (2000) andArbabi-Bidgoli & Muller (2002). It uses a 3003 grid andincludes voids with a minimum effective radius of 3 h−1 Mpc.In the following, this void finder and its results are referredto as Muller.


Figure 1. A slice of thickness 5 h−1 Mpc through the centre of the region extracted from the Millennium simulation. The image showsthe dark matter distribution in the central 40 h−1 Mpc region. Void galaxies (within any void, not just the largest one) are superimposedon the dark matter distribution as blue circles. The top left and top centre panels show only the dark matter distribution and darkmatter plus all galaxies in the slice, respectively. The other panels show the locations of the largest void (with dark matter particles insidethe void marked green), its centre (red circle), and all void galaxies found by Brunino (top right), Colberg (second row, left column),Fairall (second row, centre), Foster (second row, right column), Gottlober (third row, left column), Hahn/Porciani (third row, centre),Hoyle/Vogeley (third row, right column), Muller (bottom, left column), Neyrinck (bottom, centre), Pearce (bottom, right column).


Author NV FFV δDM Ng δg Ng,20 δg,20 (xmax, ymax, zmax) r

[Mpc/h] [Mpc/h]

Brunino P 3 0.37 -0.78 754 -0.71 7 -0.93 (38.6, 46.8, 199.5) 16.0Colberg1 DM 21 0.92 -0.74 2258 -0.65 35 -0.85 (35.3, 41.2, 193.9) 29.9Fairall P 18 0.59 -0.73 1376 -0.67 25 -0.83 (33.0, 40.0. 200.0) 20.0Foster/Nelson P 3 0.41 -0.82 114 -0.96 0 -1.00 (36.3, 36.6, 192.4) 18.0Gottlober2 P 9 0.35 -0.77 733 -0.70 0 -1.00 (32.1, 44.0, 192.0) 16.4Hahn/Porciani1,3 DM 14 0.29 -0.73 248 -0.92 0 -1.00 (30.5, 33.6, 191.8) 17.2Hoyle/Vogeley1,2 P 4 0.84 -0.68 2166 -0.56 40 -0.79 (31.9, 47.1, 193.2) 24.6Muller2 P 24 0.58 -0.76 1469 -0.65 0 -1.00 (30.7, 42.7, 189.1) 25.6Neyrinck1,3,4 DM 29 0.32 -0.68 834 -0.63 14 -0.83 (30.3, 33.5, 194.9) 11.3Pearce DM 5 0.15 -0.90 51 -0.95 0 -1.00 (35.9, 33.8, 193.5) 11.9Platen/Weygaert1 DM 167 1.0 -0.91 18 -1.00 0 -1.00 (37.5, 36.2, 194.3) 14.3Plionis/Basilakos DM 15 0.13 -0.92 0 -1.00 0 -1.00 (37.1, 33.8, 192.7) 10.0Shandarin/Feldman DM 19 0.23 -0.88 0 -1.00 0 -1.00 (31.5, 41.1, 192.7) 17.1

Table 2. An overview of some of the main results of this study: for each void finder, we give the total number of voids, NV , in thevolume considered here, the volume filling fraction, FFV , the average dark matter overdensity, δDM, of the voids, the total number ofgalaxies, Ng , found in voids, the average galaxy overdensity δg, the number of galaxies brighter than mB = −20, Ng,20, found in voids,the average galaxy overdensity using only galaxies brighter than mB = −20, δg20, and positions of the centres of the largest void andtheir radii. We also classify the void finders into those using the dark matter (smoothed or not – DM) and those using points (galaxiesor haloes – P). Notes: 1 the voids are non–spherical, so the quoted radius is an approximation, assuming a spherical void. 2 using theB < −20 galaxy sample. 3 the quoted centre of the void is actually the position of lowest density. 4 9308 voids found; of them 2362, 525,164, 64, 29, 13, and 5 exceed 1 through 7σ probability thresholds, respectively. We use the 5σ results for comparisons.

Figure 2. Same as and continued from Figure 1. Platen/Weygaert (left column), Plionis/Basilakos (centre), and Shandarin/Feldman(right column). Note that both Plionis/Basilakos and Shandarin/Feldman find no void galaxies.

4 RESULTS – COMPARISON

4.1 Basic numbers

In Table 2 we provide an overview of the results obtainedwith the different void finders. In particular, for each voidfinder, we list the total number of voids, NV , the volumefilling fraction, FFV , the average dark matter overdensity,δDM, of the voids, the total number of galaxies, Ng , found invoids, the corresponding average galaxy overdensity, δg, thenumber of galaxies brighter than mB = −20, Ng,20, foundin voids, the corresponding average galaxy overdensity usingonly those galaxies, δg20, and positions of the centres of thelargest void and their radii.

When comparing these numbers it is important to keepthe differences in the void finders in mind. For example, somevoid finders construct strictly spherical voids, whereas othersbuild larger ones out of spherical proto–voids. In addition,there are differences in the spatial resolutions. The numbers

of voids found in the volume thus can be expected to bedifferent, and they should merely be treated as illustrativequantities.

If the different results strictly reflected the density fieldin the simulation, that is if all the void samples were centredon the most underdense regions and then extended out tohigher density regions, there would be a simple relationshipbetween the volume filling fraction FFV and the averagedark matter overdensity δDM. To a certain degree such a cor-relation does exist. For example, the Pearce voids are centredon the particles with the lowest local densities and are cutoff at an overdensity of δ = −0.9, whereas Colberg voids areconstructed around proto–voids with δ = −0.8. This resultsin a much lower value of FFV for Pearce, whereas Colberg’svoids fill almost the entire volume3.

3 Recall that the subvolume studied here has a mean overdensityof δ = −0.28. So Colberg’s result is not all that surprising given


Figure 3. Radially averaged dark matter density profiles of the largest void in each of the void catalogues found by the groups involvedin the study. For each void finder the profile extends out to the largest radius that can be studied, given the size of the volume. See maintext for more details.

A more detailed examination of Table 2 reveals thekey difference between the void finders we have employed:they effectively target different mean overdensities. Thosewhich correspond to a low mean overdensity (Pearce,Platen/Weygaert, Plionis/Basilakos & Shandarin, all withδ ∼ −0.9) naturally contain very few galaxies as they pickout the deepest parts of the voids. At the other extreme,those finders which effectively employ higher overdensitythresholds (for instance Colberg and Hoyle/Vogeley withδ ∼ −0.7) pick out much larger regions and naturally en-close far more galaxies. There is nothing intrinsically wrongwith different void finders targeting different overdensities,in fact in some sense pretty much the whole region could beclassed as a “void”, in that it has far less dark matter thanexpected and consequently is depleted of galaxies. As a re-sult secondary characteristics such as the void radius or thenumber density of void galaxies need to be calibrated againstthis effective threshold before techniques can be comparedin detail. Such a study is beyond the scope of this paper butshould be borne in mind when examining such measures asthe largest void in any particular dataset.

Given that not all void finders are density–based, there

the procedure it uses and the fact that the whole region is quiteunderdense.

also is no direct relationship between FFV and the numberof galaxies inside the voids, Ng. There is a clear difference inNg between the different models, with Plionis/Basilakos andShandarin/Feldman finding no void galaxies whatsoever4,and the most extreme cases with several thousand voidgalaxies. Given the fact that the volume studied here hasa mean overdensity of δ = −0.28 finding lots of void galax-ies is maybe not all that surprising – provided one is happywith the existence of such objects. The number of galax-ies brighter than mB = −20, Ng,20, is either zero or verysmall for all void finders. This is an important agreementfor void finders which accept the existence of galaxies invoids: The overdensity of such galaxies, δg20, is smaller thanabout −0.8, regardless of how voids are found.

There are also interesting agreements for quite differentvoid finders. For example, Plionis/Basilakos and Pearce findvery similar results (FFV , δDM, and especially the positionof the largest void, but not Ng).

In Table 2, we also give the position of the centre ofthe largest void and its radius, as provided by the differ-ent groups. Note that some void finders build non–spherical

4 Note, though that for Plionis/Basilakos a change of the pdffraction, below which “void cells” are considered, to the lowest30% of the pdf, results in finding void galaxies.


Full box

all galaxies

PearceFairallBrunino

ColbergHoyle/Vogeley

Gottloeber

NeyrinckFoster/Nelson

Mueller

Platen/WeygaertHahn/Porciani

Figure 4. Space density of galaxies (h3/Mpc3/mag) as a function of dust corrected MB for galaxies in the volume under considerationand in the catalogues of those void finders which identify galaxies inside voids. For purposes of comparison, the luminosity function ofthe full simulation volume is also given. Each void finder luminosity function is corrected for the volume occupied by the relevant void

sample.

voids, so the quoted radius merely reflects the total size ofthe void. While all the finders indeed locate a large voidwithin the central region it is perhaps a little surprising thatsome centres are not within the central structure that is soclearly visible in the top left panel. This is in fact anotherconsequence of the varying density thresholds employed inthat those finders with effectively lower thresholds rely onlarger scale structures than those that employ very low den-sity thresholds. In addition some methods (such as those ofBrunino and Gottlober) find several voids of nearly equal sizein this region, as evidenced by the number of marked bluegalaxies on Figure 1 that are not within the marked greenvoid. The key point is that the filamentary structures visiblein the top left panel of Figure 1 are not very massive. Againit is clear that void sizes depend quite strongly on how voidsare found, so one has to be very careful about using voidsizes to make statements about large–scale structure.

In Figures 1 and 2, we show void galaxies found by thedifferent groups. As noted above, the top left and top centrepanels of Figure 1 give only the dark matter distributionand the dark matter plus all model galaxies, respectively.All other panels superimpose all the recovered void galaxieson top of the dark matter distribution. In addition, for each

group, we also show the largest void in green. The void centreis marked with a large red dot.

As is clearly visible, there are quite large variations be-tween the different groups, a direct consequence of the widevariety of techniques and limits employed. Hopefully thesefigures shed some light on the question of what each groupactually means when they refer to a “void” and illustratethe inherent difficulty of comparing results obtained usingdifferent void finders. Individually the results of each groupmake perfect sense, when seen in the light of how voids areidentified. For example, the Pearce voids are some of themost underdense spheres in the volume, centred on the par-ticles with the lowest density. Conversely, at first glance,the Colberg void doesn’t appear void at all and spans theentire figure, but this is a natural consequence of the verylow density of the entire region.

So unless agreement has been achieved on how to definewhat a void really is – or should be – it is not straightforwardto argue which void finder does the best job, at least whencomparing images.


Figure 5. Distributions of the local densities of the galaxies in the results of those void finders that identity void galaxies. The localdensity is expressed via r14, which for each galaxy gives the radius of the sphere around the galaxy that contains 1014 h−1 M⊙. Forcomparison purposes, the distribution of the full galaxy sample is also shown.

4.2 Void Density profiles

Despite the differences in the void–finding methods em-ployed in earlier studies, there has been broad agreement ontwo facts, namely that voids are very empty in their centresand that they have very sharp edges (see, for example, Ben-son et al. 2003, Colberg et al. 2005, or Patiri et al. 2006b).Given the differences in the void finders, “very empty” mightmean different things. It might mean that the voids are lit-erally empty of the objects used as data – such as galaxiesbelow some given luminosity, say – or that voids do containsome material (for example dark matter), but very little ofit.

With the large variety of void finders used here, it isan interesting and important point to study the internalstructure of a void. With the volume under considerationrelatively small and underdense, most of the void findersfind one very large void, at about the same location. We arethus limited to studying the structure of the largest void ineach catalogue.

Figure 3 shows the radially averaged enclosed dark mat-ter density as a function of radius for each of the catalogues,using the void centres given in Table 2 and shown visuallyon Figures 1 and 2 as a red point. It is quite important tonote that such a radial average is not ideal for void findersthat produce non–spherical voids. Also, for each void finder

the profile extends out to the largest radius that can bestudied, given the size of the volume, so only the profiles ofvoids that lie close to the centre of the volume extend be-yond 25h−1 Mpc. Note that these radially averaged densityprofiles cannot be easily compared with the average over-densities quoted in Table 2. The values quoted in Table 2were computed using only the total void volume. However,radially averaging as in Figure 3 for voids that are not per-fectly spherical will include material that does not lie insidea void.

It appears that the void finders fall into three broadcategories, namely those which have central densities wellbelow δ = −0.9 (Foster/Nelson, Hahn/Porciani, Neyrinck,Platen/Weygaert, Plionis/Basilakos, Pearce), those withmuch higher, flat, central densities (Brunino, Gottlober,Hoyle/Vogeley, Muller) and a third set with intermedi-ate central densities (Colberg, Fairall, Shandarin/Feldman).Void finders with very low central densities all use the darkmatter density field in order to identify voids in combinationwith a low effective overdensity threshold which restricts thesize of the voids. The void finders that use (model) galaxiesor haloes all have somewhat higher central densities, andmuch flatter central profiles. This effect is partly due to theinclusion of small haloes near the void centres as well as thedifficulty of defining a void from a sparse sample of points.


Nevertheless it is clear that voids selected using the sparsetracers available from galaxies or haloes typically have cen-tral overdensities around δ = −0.85 whereas those selectedfrom the richer dark matter distribution have typically lowercentral density limits. Figure 3 further illustrates the role ofthe effective overdensity threshold driving void choice: inthe central region the method of Colberg does not recover aparticularly deep void, however, between 15 and 20h−1Mpcthis method has found the most underdense region of all thefinders.

Up to a radius of around 15h−1 Mpc, the largest voidin each catalogue has an average density of δ ≈ −0.85 andat larger radii the radially averaged densities are all rising.However the entire volume studied here has a mean δ =−0.28 so none of the voids runs into the very steep edges seenin earlier work as we are still well below the mean cosmicdensity.

Despite the differences in the central densities, we canconclude that regardless of how voids are found, their inte-riors are very underdense and they contain mean densitiesof between 5% and 20% of the cosmic mean. The centralregions of voids also tend to have a rather flat profiles whichmeans that regardless of how voids are found in observa-tional surveys, follow-up work of their interiors – such as, forexample, searches for hydrogen (see, for example, Giovanelliet al. 2005) – should expect very low densities of material,provided, of course, that the current model of structure for-mation used in the simulation is correct.

4.3 Luminosity Function

Figure 4 shows the luminosity functions of galaxies in theentire Millennium simulation (solid red line), that of thevolume under consideration (solid black line) and in each ofthe catalogues of those void finders which identify galaxiesinside voids, colour coded as shown on the figure.

We present this plot mostly for illustrative purposes,since the volume under consideration here is quite small. Thekey difference between the full simulation and our selectedsubvolume is that the galaxy formation efficiency across thisentire region has been suppressed. In the full Millennium vol-ume there are 7,151,282 galaxies with MB between -16 and-22. If the central 40h−1 Mpc of our subvolume was a ran-dom section of the full box you would expect to find 3,661galaxies. In practice our region has 707, or less than 1

5of

the expected number. As well as this overall normalisation,compared with the full volume of the simulation, the lumi-nosity function of the subvolume is very slightly steeper atthe faint end and is deficient in bright galaxies.– as men-tioned before, the subvolume is underdense, so we do notexpect to find many bright galaxies.

The luminosity functions of the samples that con-tain significant numbers of galaxies (with the exception ofFairall) show an even greater deficiency of bright galaxies, asevidenced earlier by the very low overdensity of bright galax-ies in voids (Section 4.1). Although it is difficult to tell, itlooks as if at the fainter end, the luminosity functions of thevoid samples all are just very slightly steeper than the sub-volume one’s and slightly steeper than the full simulationvolume one’s. This could be seen as a trend towards themost isolated galaxies being fainter than expected, as wouldbe suggested from theoretical arguments. Those galaxies re-

siding in the most underdense regions (although there aren’tvery many) are certainly faint. The limit of this effect, nogalaxies in the voids, is achieved by two finders, those ofPlionis/Basilakos and Shandarin/Feldman.

4.4 Void galaxies and local environments

Given that we are interested in comparing results from dif-ferent void finders, it is worthwhile to look at which galax-ies void finders identify as belonging to a void. Apart froma galaxy’s brightness, its environment, expressed throughsome measure of the local density, provides a useful descrip-tor. In order to quantify the local density, for each galaxy wecompute r14, the radius of the sphere that contains a massof 1014 h−1 M⊙, roughly the mass of a small galaxy cluster.

In Figure 5, we show the distribution of the values ofr14, for both the complete subsample and the individual voidgalaxy sets. Large (small) values of r14 correspond to regionsof low (high) density. The distribution reflects the fact thatthe subvolume considered here is underdense, since mostgalaxies reside in the low–density part of the distribution.

One would naively expect that void finders would pickup the galaxies in the lowest density regions first and thenmove towards the higher density regions. However, while thisis true for some of the void finders, it is not true for all ofthem. This fact should be an important criterion for futurediscussions of void finders: if a void finder locates galaxiesinside voids, should these be those in the most underdenseenvironments?

Interestingly enough, the purely visual Fairall void–finding results in a distribution that is quite similar to Col-berg’s, and also to Neyrinck’s and Muller’s. Given the largedifferences in the methods these similarities are quite inter-esting, and they merit to be taken into account in futurediscussions of how to find voids.

5 SUMMARY AND DISCUSSION

This study represents the first systematic study of thir-teen void finders, all of which have been used over the pastdecade to study voids, using the same data set to compareresults. For the data we used real–space coordinates of par-ticles, haloes, and semi–analytical model galaxies (Crotonet al. 2005) from a subvolume of the Millennium simulation(Springel et al. 2005). The goal of this paper was not to argueabout the best way to define or identify voids. Instead, weaimed at allowing the reader to understand the differencesbetween the methods to allow easier comparison of studiesof voids in the literature.

As outlined in Table 1, the void finders in this studyrange from studies of the smoothed dark matter density fieldto identifying empty spheres in the distribution of modelgalaxies, the latter either using sophisticated algorithms orsimply the human eye. Given the vastly different assump-tion of what a void actually is, it is not surprising to seelarge differences between some of the void finders. However,there are also some quite encouraging agreements betweenmethods that are quite different.

Not surprisingly, the different methods result in a largespread in basic numbers such as the number of voids, the sizeof the largest voids (see Table 2), or their basic appearance


(see Figures 1 and 2). We caution against putting too muchemphasis on this fact. If one void finder constructs sphericalvoids with mean overdensities of δ = −0.9 and another onebuilds large, irregularly shaped voids from spherical proto–voids in a distribution of galaxies, then the numbers andsizes of voids can be expected to be quite different. Likewise,the fraction of volume filled by the voids will be different.Regardless of these differences, it is quite interesting to seethat the locations of the largest voids found by most of thegroups agree quite well with each other. The eye finds a largevoid in the centre of the region studied, and the void findersdo the same!

For a more detailed comparison the effective overdensityproves to be most interesting. Here, the spread is not quite asextreme as expected (see the values of δDM in Table 2), andthe agreement in the overdensities of bright galaxies is quiteimpressive. The void finders in this study agree that thereshould be no or just a very small number of bright galaxiesin voids. In other words, regardless of how one defines voids,there are almost no bright galaxies in them.

As Section 4.2 shows, the differences in the (radiallyaveraged) density profiles of the largest void are also notvery large, with the void centres containing only between5% and 20% of the mean density. This means that regard-less of how voids are found, their centres contain very littlemass – unless, of course, our model of cosmic structure for-mation, which forms the basis of the simulation, is wrong.With searches for HI emission in voids under way (see Gio-vanelli et al. 2005 or Basilakos et al. 2007), there shouldsoon exist additional data points, which makes it all themore pressing to move towards a more unified picture ofvoids.

As just mentioned, voids contain very few bright galax-ies, and they contain relatively more dim galaxies, somethingthat those void finders that identify void galaxies appearto agree on, too (see Section 4.3). Given the small numberstatistics in our sample, it is impossible to make strongercomments about this. What appears clear, though, is thatthis is an important topic to study, both observationally andtheoretically, in particular since current models of galaxyformation and evolution have to account for the observedrelation. For these studies to be successful, more commonground is needed as far as defining and finding voids is con-cerned.

We hope that this paper will trigger more detailedfollow–up studies to work towards a more unified view ofhow to define and find voids. We believe that studies likethis one, which make use of high–resolution simulations oflarge–scale structure, provide invaluable tools to this end,since they contain full information about the distributionof model galaxies and of the underlying density field. In theend, the model could then still be entirely wrong – a possibil-ity that, in the light of the recent development of a standardcosmological model, appears to be somewhat unlikely – butit will still be able to provide a sound basis for calibrationsof methods and ideas. This point is of particular interestsince observationally (at present) only galaxies can be usedto find voids. The distribution of galaxies is much harderto model, though, than the cosmic density field – the lattercan be described quite well using linear theory. For studies ofvoids to be useful, a link needs to be forged between theoryand observation. We hope that this work will provide a basis

for resolving this situation. Ultimately, as one of the mostextreme cosmic environments, voids possess the potential toconstrain models of galaxy formation. But for that to be thecase, we need to agree on what they really are and how tofind them.

ACKNOWLEDGEMENTS

This work was initiated at the Aspen Center for Physics’workshop on Cosmic Voids (June 2006) and at The RoyalNetherlands Academy of Arts and Sciences Colloquium onCosmic Voids (December 2006). We thank both institutionsfor their generous and valuable support of this area of re-search. We also thank the Virgo Supercomputing Consor-tium and the Lorentz Center in Leiden for their hospitalityduring the Winter 2007 Virgo Meeting.

The Millennium simulation used in this paper wascarried out by the Virgo Supercomputing Consortium(http://www.virgo.dur.ac.uk) at the Computing Centre ofthe Max-Planck Society in Garching. We thank the DarrenCroton for making the galaxy catalogue publicly available.Much of the preparation and pre–analysis required for thiswork was carried out at the Nottingham HPC facility.

J. Colberg is indebted to Carlos Frenk, Adrian Jenk-ins, and Lydia Heck for their support and help withDurham computing facilities, where part of this workwas completed. C. Foster would like to thank NSERC(Canada) for support in the form of a graduate scholar-ship. L. Nelson would like to thank the Canada ResearchChairs Program and NSERC for financial support. Fig-ure 1 was produced using Nick Gnedin’s IFRIT software(http://home.fnal.gov/∼gnedin/IFRIT).

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